Operations on [[Vectors]] ([[Vector spaces]]). # Addition & Subtraction Suppose $\underline{v} = \begin{pmatrix} a_{1} \\ \vdots \\ a_{n} \end{pmatrix}, \quad \underline{w} = \begin{pmatrix} b_{1} \\ \vdots \\ b_{n} \end{pmatrix} \quad \in \mathbb{F}^{n}$ > [!info] Remark > It is usual to think of this addition of vectors as completing a parallelogram with a vertex at the origin and v and w as two adjacent sides there. > # Scalar Multiplication > [!Remark] > $\{ \lambda \underline{v} \mid \lambda \in \mathbb{R} \}$ is the line through $\underline{v}$. > # Properties > [!Lemma] > Let $\underline{v}, \underline{w}$ be vectors and $\lambda, \mu$ be scalars. Then > 1. $\underline{v} + \underline{w} = \underline{w}+ \underline{v}$, [[Commutativity]] of vector addition > 2. $\lambda(\underline{v} + \underline{w}) = \lambda \underline{v} + \lambda \underline{w}$, scalar multiplication is [[Distributivity|distributive]] over vector addition. > 3. $\lambda(\mu \underline{v}) = (\lambda \mu) \underline{v}$, [[Associativity]] of scalar multiplication > 4. $0 \underline{v}= \underline{0}$, $1\underline{v} = \underline{v}$, $(-1)\underline{v} = -\underline{v}$, $\underline{v}+\underline{v}=2\underline{v}$ and $\underline{v} - \underline{v} = \underline{v} + (-\underline{v}) = \underline{0}$. > [!Proof] > Consider each component of a vector separately. > 1. By definition, the $i$-th component of $\underline{v} +\underline{w}=a_{i}+b_{i}$ > $i$-th component of $\underline{w}+\underline{v}=b_{i}+a_{i}=a_{i}+b_{i}$ > so $\underline{v}+\underline{w}=\underline{w}+\underline{v}$. > 2. $i$-th component of $\lambda(\underline{v}+\underline{w})=\lambda(a_{i}+b_{i})=\lambda a_{i}+\lambda b_{i}$ which is the $i$-th component of $\lambda\underline{v}+\lambda \underline{w}$ so $(2)$ holds. > 3. etc.